Models of low-speed flow for near-critical fluids with gravitational and capillary effects

Authors:
D. L. Denny and R. L. Pego

Journal:
Quart. Appl. Math. **58** (2000), 103-125

MSC:
Primary 76N10; Secondary 35Q35, 76D45

DOI:
https://doi.org/10.1090/qam/1738560

MathSciNet review:
MR1738560

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study low-speed flows of a highly compressible, single-phase fluid in the presence of gravity, for example, in a regime appropriate for modeling recent space-shuttle experiments on fluids near the liquid-vapor critical point. In the equations of motion, we include forces due to capillary stresses that arise from a contribution made by strong density gradients to the free energy. We derive formally simplified sets of equations in a low-speed limit analogous to the zero Mach number limit in combustion theory.

**[1]**D. M. Anderson, G. B. McFadden, and A. A. Wheeler,*Diffuse-interface methods in fluid mechanics*, Annual review of fluid mechanics, Vol. 30, Annu. Rev. Fluid Mech., vol. 30, Annual Reviews, Palo Alto, CA, 1998, pp. 139–165. MR**1609626**, https://doi.org/10.1146/annurev.fluid.30.1.139**[2]**G. K. Batchelor,*The conditions for dynamical similarity of motions of a frictionless perfect-gas atmosphere*, Quart. J. Roy. Meteo. Soc.**79**, 224-235 (1953)**[3]**R. F. Berg,*Thermal equilibration near the critical point: Effects due to three dimensions and gravity*, Phys. Rev. E**48**, 1799-1805 (1993)**[4]**H. Boukari, M. E. Briggs, J. N. Shaumeyer, and R. W. Gammon,*Critical speeding up observed*, Phys. Rev. Lett.**65**, 2654-2657 (1990)**[5]**H. Boukari, R. Pego, and R. W. Gammon,*Calculation of the dynamics of gravity-induced density profiles near a liquid-vapor critical point*, Phys. Rev. E**52**, 1614-1625 (1995)**[6]**H. Boukari, J. N. Shaumeyer, M. E. Briggs, and R. W. Gammon,*Critical speeding up in pure fluids*, Phys. Rev. A**41**, 2260-2263 (1990)**[7]**Alexandre J. Chorin and Jerrold E. Marsden,*A mathematical introduction to fluid mechanics*, 3rd ed., Texts in Applied Mathematics, vol. 4, Springer-Verlag, New York, 1993. MR**1218879****[8]**D. L. Denny and R. L. Pego,*Solutions for a model of low-speed flow for highly compressible fluids with capillary effects*, in preparation**[9]**J. E. Dunn and J. Serrin,*On the thermomechanics of interstitial working*, Arch. Rational Mech. Anal.**88**(1985), no. 2, 95–133. MR**775366**, https://doi.org/10.1007/BF00250907**[10]**D. Durran,*Improving the anelastic approximation*, J. Atmos. Sci.**46**, 1453-1461 (1989)**[11]**J. A. Dutton and G. H. Fichtl,*Approximate equations of motion for gases and liquids*, J. Atmos. Sci.**26**, 241-254 (1969)**[12]**P. Embid,*Well-posedness of the nonlinear equations for zero Mach number combustion*, Comm. Partial Differential Equations**12**(1987), no. 11, 1227–1283. MR**888460**, https://doi.org/10.1080/03605308708820526**[13]**R. Gammon, personal communication. Also see the ZENO home page at the URL http://roissy.umd.edu/. The experimental design is described at the URL http://roissy. umd.edu/usmp3/reminder.html.**[14]**C. Ikier, H. Klein, and D. Woermann,*Optical observation of the gas/liquid, phase transition in near-critical under reduced gravity*, J. Coll. Internat. Sci.**178**, 368-370 (1996)**[15]**C. Ikier, H. Klein, and D. Woermann,*Density equilibration in a near-critical fluid under reduced gravity*, Ber. Bunsenges. Phys. Chem.**100**(8), 1308-1311 (1996)**[16]**Sergiu Klainerman and Andrew Majda,*Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids*, Comm. Pure Appl. Math.**34**(1981), no. 4, 481–524. MR**615627**, https://doi.org/10.1002/cpa.3160340405**[17]**A. B. Kogan, F. Zhong, and H. Meyer,*Dynamics of density equilibration near the liquid-vapor critical point of He*--3, Czechoslovak J. Phys.**46**, Suppl. 1, 71-72 (1996)**[18]**L. D. Landau and E. M. Lifshitz,*Course of theoretical physics. Vol. 6*, 2nd ed., Pergamon Press, Oxford, 1987. Fluid mechanics; Translated from the third Russian edition by J. B. Sykes and W. H. Reid. MR**961259****[19]**F. B. Lipps and R. S. Hemler,*A scale analysis of deep moist convection and some related numerical calculations*, J. Atmos. Sci.**39**, 2192-2210 (1982)**[20]**A. Majda,*Compressible fluid flow and systems of conservation laws in several space variables*, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR**748308****[21]**A. Majda and J. Sethian,*The derivation and numerical solution of the equations for zero Mach number combustion*, Combust. Sci. Tech.**42**, 185-205 (1985)**[22]**M. R. Moldover, J. V. Sengers, R. W. Gammon, and J. R. Hocken,*Gravity effects in fluids near the gas-liquid critical point*, Rev. Mod. Phys.**51**, 79-99 (1979)**[23]**Y. Ogura and N. A. Phillips,*Scale analysis of deep and shallow convection in the atmosphere*, J. Atmos. Sci.**19**, 173-179 (1962)**[24]**A. Onuki and R. A. Ferrell,*Adiabatic heating effect near the gas-liquid critical point*, Physica A**164**, 245-264 (1990)**[25]**A. Onuki, H. Hao, and R. A. Ferrell,*Fast adiabatic equilibration in a single-component fluid near the liquid-vapor critical point*, Phys. Review A**41**, 2256-2259 (1990)**[26]**V. A. Rabinovich,*Thermophysical Properties of Neon, Argon, Krypton, and Xenon*, Hemisphere Publishing Corp., New York, 1988**[27]**R. G. Rehm and H. R. Baum,*The equations of motion for thermally driven, buoyant flows*, J. Res. Natl. Bur. Stand.**83**, 297-308 (1973)**[28]**J. S. Rowlinson and B. Widom,*Molecular Theory of Capillarity*, Clarendon Press, Oxford, 1982**[29]**J. V. Sengers,*Transport properties of fluids near critical points*, Internat. J. Thermophysics**6**, 203-232 (1985)**[30]**J. V. Sengers, R. S. Basu, and J. M. H. Levelt Sengers,*Representative equations for the thermodynamic and transport properties of fluids near the gas-liquid critical point*, NASA Contractor Report 3424, 1981**[31]**H. L. Swinney and D. L. Henry,*Dynamics of fluids near the critical point: Decay rate of order-parameter fluctuations*, Phys. Rev. A**8**, 2586-2617 (1973), and references therein**[32]**R. Wilhelmson and Y. Ogura,*The pressure perturbation and the numerical modeling of a cloud*, J. Atmos. Sci.**29**, 1295-1307 (1972)**[33]**Chia Shun Yih,*Stratified flows*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Second edition of Dynamics of nonhomogeneous fluids. MR**569474****[34]**B. Zappoli and P. Carles,*The thermo-acoustic nature of the critical speeding up*, Eur. J. Mech. B/Fluids**14**, 41-65 (1995)**[35]**B. Zappoli, S. Amiroudine, P. Carlès, and J. Ouazzani,*Thermoacoustic and buoyancy-driven transport in a square side heated cavity filled with a near critical fluid*, J. Fluid Mech.**316**, 53-72 (1996)**[36]**F. Zhong and H. Meyer,*Density equilibration near the liquid-vapor critical point of a pure fluid: Single phase*, Phys. Rev. E**51**3223-3241 (1995)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
76N10,
35Q35,
76D45

Retrieve articles in all journals with MSC: 76N10, 35Q35, 76D45

Additional Information

DOI:
https://doi.org/10.1090/qam/1738560

Article copyright:
© Copyright 2000
American Mathematical Society